Decision making under uncertainty is a hard subject. It is hard because there is no single criterion that is universally accepted by decision makers. For example, when the probability of the ‘state’ of the world is hard to evaluate, a risk averter (a person who averts risks) may make decision based on the worst scenario, whereas a risk seeker would like to bet on the optimistic scenario. Even if the worst scenario is used as the decision criterion, the decision will depend on which side of the coin is checked. At one side of the coin is the payoff, at the other side, regret. If the worst payoff is focused, the decision maker might choose among the alternative actions one that has the maximum minimum payoff, i.e., the maximin criterion. Otherwise, she might choose the action that minimizes the maximum regret, and hence she exercises the minimax criterion.
Similarly, when the probability of the state of the world is given, the decision criteria are not unique. For instance, one might make decision completely based on the maximum likelihood scenario. Nevertheless, the most often used decision criterion is the so-called maximized expected value criterion. Based on this criterion, a decision maker chooses the action of the maximum expected value. Suppose that the ‘world’ of the decision problem has only a finite number of states, and that there is a known probability pi associated with each state i . Then the expected value is calculated as
EV = p1×v1+…+pn×vn
where vi is the value of the outcome when the state is in state i for this action. Here the value is used as a synonym of utility.
As shown in the expression above, the expected value criterion involves two major terms: probability and utility. Unfortunately, neither of the terms is easy to understand.
A simple question to ask is: Why should the maximized expected value criterion be considered rational? In other words, why is it rational for a decision maker to maximize the expected value?
There are two different arguments for the expected value principle. The first argument is based on the law of large numbers (LLN) in probability. That is, it is a rational criterion because in the long run the decision maker will be better off if s/he maximizes expected value. The second argument takes an axiomatic approach, aiming at deriving the expected value criterion from a few fundamental well-accepted facts (axioms) for rational decision making.
However, the LLN argument has been refuted by many prominent decision theorists from several aspects. Keynes famously objected to the LNN by stating: “In the long run we are all dead”. He suggested that no real-life decision maker will ever face any decision an infinite number of times. Therefore, mathematical facts about what would happen after an infinite number of repetitions are therefore of little normative relevance. Just on the contrary the real-life decisions are often of one-shot nature.